Search

Decays $B\to DP\ell^+\nu_\ell~(\ell=e,\mu,\tau)$ with the non-resonance, the charmed vector resonances, the charmed scalar resonances and the charmed tensor resonances are calculated by using the SU(3) flavor symmetry. Firstly, the decay amplitudes of different modes are related by the SU(3) flavor symmetry. Then, relevant experiential data are used to constrain nonperturbative coefficients in the non-resonant and various resonant $B\to DP\ell^+\nu_\ell$ decays. Finally, using the constrained nonperturbative coefficients, the branching ratios of not-yet-measured $B\to D^*P\ell^+\nu_\ell$ decays with the non-resonant and various charmed resonant contributions are predicted. Many branching ratios are predicted for the first time. We find that $B\to D\eta'\ell^+\nu_\ell, B_s\to D_s\eta'\ell^+\nu_\ell$ decays only receive the non-resonant contributions, $B\to D_sK\ell^+\nu_\ell$, $B_s\to DK\ell^+\nu_\ell$, $B\to D\eta\ell^+\nu_\ell$ and $B_s\to D_s\eta\ell^+\nu_\ell$ decays receive both non-resonant and charmed tensor resonant contributions, $B^+\to D^-\pi^+\ell^+\nu_\ell$ decays receive the non-resonant, the charmed scalar resonant and the charmed tensor resonant contributions, and other $B\to D\pi\ell^+\nu_\ell$ decays receive all four kinds of contributions. These results can be used to test the SU(3) flavor symmetry approach in the four-body semileptonic decays by the future LHCb and Belle-II experiments., Comment: 16 pages. arXiv admin note: text overlap with arXiv:2403.14929

Motivated by the significant experimental progress in probing semileptonic decays $D\to P_1P_2\ell^+\nu_\ell~(\ell=\mu,e)$, we analyze the branching ratios of the $D\to P_1P_2\ell^+\nu_\ell $ decays with the non-resonant, the light scalar meson resonant and the vector meson resonant contributions in this work. We obtain the hadronic amplitude relations between different decay modes by the SU(3) flavor analysis, and then predict relevant branching ratios of the $D\to P_1P_2\ell^+\nu_\ell$ decays by the present experimental data with $2 \sigma$ errors. Most of our predicted branching ratios are consistent with present experimental data within $2\sigma$ error bars, and others are consistent with the data within $3\sigma$ error bars. We find that some branching ratios of the non-resonant decays are on the order of $\mathcal{O}(10^{-3}-10^{-4})$, the vector meson resonant contributions are dominant in many decays, the non-resonant, the vector meson resonant and the scalar resonant contributions are all important in the $D^0\to\eta\pi^-\ell^+\nu_\ell$ decays, and both the non-resonant and the scalar resonant contributions are important in the $D^0\to K^-K^0\ell^+\nu_\ell,\eta'\pi^-\ell^+\nu_\ell$ and $D^+\to \overline{K}^0K^0\ell^+\nu_\ell,\pi^0\pi^0\ell^+\nu_\ell,\eta\pi^0\ell^+\nu_\ell,\eta'\pi^0\ell^+\nu_\ell$ decays. According to our predictions, many decay modes could be observed in the experiments like BESIII, LHCb and BelleII, and some decay modes might be measured in these experiments in near future., Comment: 24 pages, 1 figure

Within the Standard Model, we investigate the CP violations and the $K_S^0-K_L^0$ asymmetries in $D^{\pm}\rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S,L}^{0}\pi^0 \pi^{\pm}$ decays basing on the factorization-assisted topological-amplitude (FAT) approach and the topological amplitude (TA) approach of Cheng and Chiang [Phys. Rev. D 104, 073003 (2021).]. We find that the CP violations in these decays $A_{CP}^{K_{S,L}^0}$ can exceed the order of $10^{-3}$ in the two approaches and consist of three parts: the indirect CP violations in $K^0 -\bar{K}^0$ mixing $A_{CP,K_{S,L}^0}^{mix}$, the direct CP violations in charm decays $A_{CP,K_{S,L}^0}^{dir}$ and the new CP violation effects $A_{CP,K_{S,L}^0}^{int}$, which are induced from the interference between two tree (Cabibbo-favored and doubly Cabibbo-suppressed) amplitudes with the neutral kaon mixing. The indirect CP violations in $K^0 -\bar{K}^0$ mixing play a dominant role in $A_{CP}^{K_{S,L}^0}$, the new CP violation effects have a non-negligible contribution to $A_{CP}^{K_{S,L}^0}$. We estimate the numerical results of the $K_S^0-K_L^0$ asymmetries $R_{K_{S}-K_{L}}^{D^{\pm}}$ and find that there exist a large difference between the numerical results in the FAT approach and that of the TA approach. We also find that if ones adopt the values of the decay time parameters $t_0 = 3.0 \tau_S$ and $t_1 = 10.0 \tau_S$, the new CP violation effect $A_{CP,K_{S}^0}^{int}$ would dominate the CP violation in $D^{\pm} \rightarrow K^{\ast}(892)^{0} \pi^{\pm} + \bar{K}^{\ast }(892)^{0}\pi^{\pm}\rightarrow K_{S}^0\pi^0 \pi^{\pm}$ decays and could be observed with $6.7\times 10^{6}$ and $6.5\times 10^{6}$ $D^{\pm}$ events-times-efficiency in the FAT approach and the TA approach, respectively. Our results could be tested by the LHCb, Belle II and BESIII experiments., Comment: 38 pages

The decays $K^{\ast}(892)\rightarrow K_{S,L}^{0}\pi$ can be used to study CP violation and CPT violation. The $K^{\ast}(892)$ (hereinafter referred to as $K^{\ast}$) meson can be produced via $J/\psi$ decays at BESIII. In this paper, we study CP violation and $K_S^0-K_L^0$ asymmetry in the $J/\psi$ decays involving the $K^{\ast}$ meson in the final states and obtain the following results \begin{align} {\mathcal A}_{CP}\left(J/\psi\rightarrow K^{\ast}f_{J/\psi}\rightarrow K^0_{S}\pi f_{J/\psi}\right)=\left(3.64\pm0.04\right)\times 10^{-3},\nonumber\\ {\mathcal A}_{CP}\left(J/\psi\rightarrow K^{\ast}f_{J/\psi}\rightarrow K^0_{L}\pi f_{J/\psi}\right)=\left(-3.32\pm0.04\right)\times 10^{-3},\nonumber \end{align} and \begin{align} &R\left(J/\psi\rightarrow K^{\ast}f_{J/\psi}\rightarrow K^0_{S,L}\pi f_{J/\psi} \right)=\left(3.51\pm0.03\right)\times 10^{-3},\nonumber\\ &R\left(J/\psi\rightarrow \bar{K}^{\ast} \bar{f}_{J/\psi}\rightarrow K^0_{S,L}\pi \bar{f}_{J/\psi} \right)=\left(-3.45\pm0.03\right)\times 10^{-3}.\nonumber \end{align} Basing on two cases: the samples of $10^{10}$ and $10^{12}$ $J/\psi$ events, we calculate the expected numbers of the observed signal events on the CP violation and the $K_S^0-K_L^0$ asymmetry in the $J/\psi$ decays with $K^{\ast}$ meson in the final states, we find that the BESIII experiment may be able to unambiguously observe CP violation and $K_S^0-K_L^0$ asymmetry for each of these two cases. We study the possibility to constraint the CPT violation parameter $z$ and discuss the sensitivity for the measurement of $z$ in $J/\psi$ decays with $K^{\ast}$ meson in the final states at BESIII. The sensitivity of the measurement of $z$ depend on the measured precision of the parameters $m_L-m_S$, $\Gamma_L$, $\Gamma_S$, $p$ and $q$ and the consistence between the values of $t_0$ and $t_1$ and the event selection criteria in experiment., Comment: 23 pages. arXiv admin note: text overlap with arXiv:2107.10683

The large data sample of the $B_c$ meson collected at the LHC experiment and the HL-LHC experiment provides us the opportunity to study the $B_c$ decays and the related physics. In this paper, we investigate the effect of $K^0-\bar{K}^0$ mixing on the the branching ratios, CP violations and CPT violations in the $B_{c}^{\pm}\rightarrow B^{\pm} K_{S,L}^{0}$ decays. We find that some of the $B_c^{\pm}\rightarrow B^{\pm} K_{S,L}^0\rightarrow f_{B^{\pm}} f_{K_{S,L}^0}$ decay chains have large branching ratios, whose maximum value can exceed the order of $10^{-6}$, the minimum number of $B_c^\pm$ events times efficiency for observing the decays at three standard deviations (3$\sigma$) level is about $ 10^6$. We study the CP asymmetries in the $B_c^{\pm}\rightarrow B^{\pm} K_{S,L}^0$ decays and find that the CP asymmetries can exceed the order of $10^{-3}$, which are dominated by $K^0-\bar{K}^0$ mixing. We give the most promising processes to observe the CP violations and the ranges of the numbers of $B_c^\pm$ events-times-efficiency needed to observe the CP asymmetries at a significance of 3$\sigma$ in these decays. We investigate the possibility to constraint the CPT violation parameter $z$ in the $B_c^{\pm}\rightarrow B^{\pm} K_{S,L}^0\rightarrow f_{B^{\pm}} f_{K_{S,L}^0}$ decays and give the most promising processes to constraint the parameter $z$., Comment: 32 pages,2 figures

Motivated by recent anomalies in FCNC $b\to s\ell^+\ell^-$, we study $B_1\to B_2\ell^+\ell^-(\ell=e,\mu,\tau)$ semi-leptonic weak decays with the SU(3) flavor symmetry, where $B_{1,2}$ are the spin-1/2 baryons of single bottomed anti-triplet $T_{b3}$, single charmed anti-triplet $T_{c3}$ or light baryon octet $T_{8}$. Using the SU(3) irreducible representation approach, we first obtain the amplitude relations among different decays, and then predict the relevant not-yet measured observables of these decays. (a) We calculate the branching ratios of the $T_{b3}\to T_8 \mu^+\mu^-$ and $T_{b3}\to T_8 \tau^+\tau^-$ in the whole $q^2$ region and in the different $q^2$ bins by the measurement of $\Lambda^0_b\to \Lambda^0 \mu^+\mu^-$. Many of them are obtained for the first time. In addition, the longitudinal polarization fractions and the leptonic forward-backward asymmetries of all $T_{b3}\to T_{8}\ell^+\ell^-$ decays are very similar to each other in certain $q^2$ bin due to the SU(3) flavor symmetry. (b) We analyze the upper limits of $B(T_{c3}\to T_{8}\ell^+\ell^-)$ by using the experimental upper limits of $B(\Lambda^+_c\to p\mu^+\mu^-)$ and $B(\Lambda^+_c\to pe^+e^-)$, and find the experimental upper limit of $B(\Lambda^+_c\to p\mu^+\mu^-)$ giving effective bounds on the relevant SU(3) flavor symmetry parameters. The predictions of $B(\Xi^0_c \to \Xi^0e^+e^-)$ and $B(\Xi^0_c \to \Xi^0\mu^+\mu^-)$ will be different between the single-quark transition dominant contributions and the W-exchange dominant ones. (c) As for $T_{8}\to T'_8 \ell^+\ell^-$ decays, we analyze the single-quark transition contributions and the W-exchange contributions by using two measurements of $ B(\Xi^0\to \Lambda^0 e^+e^-)$ and $ B(\Sigma^+\to p\mu^+\mu^-)$, and give the branching ratio predictions by assuming either single-quark transition dominant contributions or the W-exchange dominant ones., Comment: 17 pages

The weak and electromagnetic radiative baryon decays of octet $T_{8}$, decuplet $T_{10}$, single charmed anti-triplet $T_{c3}$ and sextet $T_{c6}$, single heavy bottomed anti-triplet $T_{b3}$ and sextet $T_{b6}$ are investigated by using SU(3) flavor symmetry irreducible representation approach. We analyze the contributions from a single quark transition $q_1\to q_2\gamma$ and $W$ exchange transitions, and find that the amplitudes could be easily related by SU(3) flavor symmetry in the $T_{b3,b6}$ weak radiative decays, $T_{c3,c6}$ weak radiative decays, $T_{10}\to T_{8}\gamma $ weak decays, $T_{10}\to T'_{10}\gamma $ weak decays and $T_{10}\to T_{8}\gamma $ electromagnetic decays. Nevertheless, the amplitude relations are a little complex in the $T_{8}\to T'_{8}\gamma$ and $T_{8}\to T_{10}\gamma$ weak decays due to quark antisymmetry in $T_{8}$ and $W$ exchange contributions. Predictions for branching ratios of $\Lambda^{0}_b\to n\gamma$, $\Xi^{-}_b\to \Xi^-\gamma$, $\Xi^{-}_b\to \Sigma^-\gamma$, $\Xi^{0}_b\to \Sigma^0\gamma$, $\Xi^{0}_b\to \Lambda^0\gamma$, $\Xi^{0}_b\to \Xi^0\gamma$, $\Xi^{*}\to \Xi\gamma$, $\Sigma^{*0}\to \Sigma^{0}\gamma$, $\Delta^0\to n\gamma$ and $\Delta^+\to p\gamma$ are given. The results in this work can be used to test SU(3) flavor symmetry approach in the radiative baryon decays by the future experiments at BESIII, LHCb and Belle-II., Comment: 17 pages, 2 figures

The $a_0^0(980)-f_0(980)$ mixing is one of the most potential tools to learn about the nature of $a_0^0(980)$ and $f_0(980)$. Using the $f_0(980)$-$a_0^0(980)$ mixing intensity $\xi_{af}$ measured recently at BESIII, we calculate the the branching ratio of the the isospin violation decay $J/\psi \rightarrow\gamma\eta_c \rightarrow \gamma \pi^0 a_0^0(1450)\rightarrow \gamma \pi^0 a_0^0(980)f_0(500)\rightarrow \gamma \pi^0 f_0(980) f_0(500) \rightarrow \gamma \pi^0 \pi^+\pi^- \pi^+\pi^-$. The value of the branching ratio is found to be $O(10^{-6})$, which can be observed with $10^{10}$ $J/\psi$ events collected at BESIII. The narrow peak from the $f_0(980)$-$a_0^0(980)$ mixing in the $\pi^+\pi^-$ mass square spectrum can also be observed. In addition, we study the non-resonant decay $a_0^0(1450)\rightarrow f_0(980) \pi^+\pi^-(\text{non-resonant})$, which is dominated by the $a_0^0(980)$-$f_{0}(980)$ mixing. We find that the non-resonant decay $a_0^0(1450)\rightarrow f_0(980) \pi^+\pi^-$ and the decay $a_0^0(1450)\rightarrow f_0(980) f_0(500)$ can be combined to measure the mixing intensity $\xi_{af}$ in experiment. These decays are the perfect complement to the decay $\chi_{c1}\rightarrow f_{0}(980)\pi^{0}\to\pi^{+}\pi^{-}\pi^{0}$ which had been observed at BESIII, the observations of them will make the measurement of the mixing intensity $\xi_{af}$ more precisely.

Many decays of light baryons consisting of light $u,d,s$ quarks have been measured, and these measurements will help us to understand the decay properties of light baryons. In this work, we study two-body nonleptonic weak decays of light baryon octet $(T_8)$ and baryon decuplet $(T_{10})$ by the topological diagram approach (TDA) under the SU(3) flavor symmetry for the first time. We find that (1) the TDA and the SU(3) irreducible representation approach (IRA) match consistently in $T_{10}\to T^{(')}_{8,10}P_8$ ($P_8$ is the light pseudoscalar meson octet); (2) almost all relevant not-yet-measured $\mathcal{B}(T_{10}\to T_8 P_8)$ may be predicted by using three experimental data of $\mathcal{B}(\Omega^-\to \Xi^0\pi^-,\Xi^-\pi^0,\Lambda^0K^-)$, and the upper limits of $\mathcal{B}(T_{10}\to T{'}_{10}\pi^-)$ may be obtained from the experimental upper limit of $\mathcal{B}(\Omega^-\to \Xi^{*0}\pi^-)$ by both the TDA and the IRA together, nevertheless, all new predicted branching ratios are too small to br reached in current experiments; (3) $T_8\to T'_8 P_8$ decays are quite complex in terms of the TDA, and we find that W-exchange diagrams give large and even dominant contributions by using relevant experimental data and the isospin relations., Comment: 14 pages, 1 figure

The semileptonic decays and two-body nonleptonic decays of light baryon octet ($T_8$) and decuplet ($T_{10}$) consisting of light $u,d,s$ quarks are studied with the SU(3) flavor symmetry in this work. We obtain the amplitude relations between different decay modes by the SU(3) irreducible representation approach, and then predict relevant branching ratios by present experimental data within $1 \sigma$ error. We find that the predictions for all branching ratios except $\mathcal{B}(\Xi\rightarrow \Lambda^0\pi)$ and $\mathcal{B}(\Xi^*\rightarrow \Xi\pi)$ are in good agreement with present experimental data, that implies the neglected $C_+$ terms or SU(3) breaking effects might contribute at the order of a few percent in $\Xi\rightarrow \Lambda^0\pi$ and $\Xi^*\rightarrow \Xi\pi$ weak decays. We predict that $\mathcal{B}(\Xi^{-}\rightarrow \Sigma^0\mu^-\bar{\nu}_\mu)=(1.13\pm0.08)\times10^{-6}$, $\mathcal{B}(\Xi^{-}\rightarrow\Lambda^0\mu^-\bar{\nu}_\mu)=(1.58\pm0.04)\times10^{-4}$, $\mathcal{B}(\Omega^-\rightarrow\Xi^0\mu^-\bar{\nu}_\mu)=(3.7\pm1.8)\times10^{-3}$, $\mathcal{B}(\Sigma^-\rightarrow \Sigma^0e^-\bar{\nu}_e)=(1.35\pm0.28)\times10^{-10}$, $\mathcal{B}(\Xi^-\rightarrow \Xi^0e^-\bar{\nu}_e)=(4.2\pm2.4)\times10^{-10}$. We also study $T_{10}\to T_8 P_8$ weak, electromagnetic or strong decays. Some of these decay modes could be observed by the BESIII, LHCb and other experiments in the near future. Due to the very small life times of $\Sigma^0$, $\Xi^{*0,-}$, $\Sigma^{*0,-}$ and $\Delta^{0,-}$, the branching ratios of these baryon weak decays are only at the order of $\mathcal{O}(10^{-20}-10^{-13}$), which are too small to be reached by current experiments. Furthermore, the longitudinal branching ratios of $T_{8A} \to T_{8B} \ell^- \bar{\nu}_\ell~(\ell=\mu,e)$ decays are also given., Comment: 23 pages, 4 figures

Using measured branching fraction of the decay $J/\psi \rightarrow\eta Y(2175))\rightarrow\eta\phi f_0(980)\rightarrow\eta\phi\pi^+\pi^{-}$ from the BESIII experiment, we estimate branching fraction of $J/\psi \rightarrow\eta Y(2175))\rightarrow\eta\phi f_0(980)\rightarrow\eta\phi\eta\pi^{0}$ decay, which proceeds via the $f_0(980)$-$a_0^0(980)$ mixing and the $\pi^0$-$\eta$ mixing. The branching fraction is predicted to be about $O(10^{-6})$, which can be accessed with $10^{10}$ $J/\psi$ events collected at the BESIII. The decay is dominated by the contribution from $f_0(980)$-$a_0^0(980)$ mixing. We find that the interference between the amplitudes due to $f_0(980)$-$a_0^0(980)$ mixing and that due to $\pi^0$-$\eta$ mixing is destructive. The branching fraction can be decreased by about $10\%$ owing to the interference effect. We also study the $\eta\pi^0$ mass squared spectrum, and find that a narrow peak due to the $f_0(980)$-$a_0^0(980)$ mixing in the $\eta\pi^0$ mass squared spectrum should be observed. The observation of this decay in experiment will be helpful to determine the $f_0(980)$-$a_0^0(980)$ mixing intensity and get information about the structures of the light scalar mesons., Comment: 16 pages, 3 figures

Within the QCD light-cone sum rule (LCSR) approach, we investigate the transition form factors of $D\rightarrow a_0(980)$ up to the twist-3 light-cone distribution amplitudes (LCDAs) of the scalar meson $a_0(980)$ in the two-quark picture. Using these form factors, we calculate the differential decay widths and branching ratios of the $D\rightarrow a_0(980) e^+ \nu_e$ semileptonic decays. We obtain ${\mathcal B}(D^0\rightarrow a_0^- (980) e^+ \nu_e)=(4.08^{+1.37}_{-1.22})\times 10^{-4}$ and ${\mathcal B}(D^+\rightarrow a_0^0 (980) e^+ \nu_e)=(5.40^{+1.78}_{-1.59})\times 10^{-4}$. The results are sensitive to the $a_0(980)$ inner structure. These decays can be searched for at BESIII experiment, and any experimental observations will be useful to identify internal quark contents of the $a_0(980)$ meson, which will shed light on understanding theoretical models., Comment: 19 pages,5 figures

We revisit the rare leptonic decay $B_s \to \mu^+ \mu^-$ in the two-Higgs doublet models with a softly broken $Z_2$ symmetry, namely type-I, type-II, type-X and type-Y 2HDMs. We have derived the relevant full one-loop Wilson coefficients of the four 2HDMs from the recent calculation in the aligned two-Higgs doublet model by Li, Lu and Pich, which could be mapped to all the four 2HDMs for both large and small $\tan\beta$. It is found that, a new term associated with the soft $Z_2$ symmetry breaking parameter $M$ can be enhanced by $\tan^2\beta$ in the type-II 2HDM, which has not been considered in the literature. Imposing both theoretical and experimental constraints, we have renewed the bounds on the parameter spaces of the four 2HDMs. Different from our previous paper, however, we find that all the four 2HDMs give sizable and similar contributions to $\overline{\mathcal B}(B_s \to \mu^+ \mu^-)$ within the stringently restricted parameter spaces, but very tiny as regards the mass-eigenstate rate asymmetry $\mathcal A_{\Delta\Gamma}$; this makes it unfeasible to discriminate the four types of 2HDM with the correlations between the observables in $B_s \to \mu^+ \mu^-$ decay., Comment: 27 pages, 2 tables, 4 figures; v2: a brief introduction to 2HDM added, matches published version

In this paper, we perform a complete one-loop computation of the short-distance Wilson coefficients for $B_{s,d}-\bar{B}_{s,d}$ mixings and $B_{s,d}\to\ell^+\ell^-$ decays in the Manohar-Wise model, which extends the SM scalar sector by a colour-octet and weak-doublet scalar. Based on these calculations, combined constraints on the model parameters are derived from the current flavour data, including the $B_{s,d}-{\bar{B}}_{s,d}$ mixings, $B_{s,d}\to\mu^+\mu^-$, $B\to X_s\gamma$, $B\to K^{\ast}\gamma$, $B\to\rho\gamma$, and $Z\to b \bar{b}$ decays. The future sensitivity to the model is also explored in the observables achievable with $50~{\rm fb}^{-1}$ of LHCb and $50~{\rm ab}^{-1}$ of Belle-II data. We find that the Manohar-Wise model could explain the current data, especially when the couplings of the charged colour-octet scalars to quarks are complex, with the resulting $\chi^2_{\rm min}$ being significantly smaller than that of the SM. Finally, we investigate correlations among the two isospin asymmetries $\Delta(K^*\gamma)$ and $\Delta(\rho\gamma)$, as well as the averaged time-integrated branching ratios $\bar{{\mathcal B}}(B_{s,d}\to \mu^+\mu^-)$; some of them are found to be quite strong and could, therefore, provide further insights into the model, once more precise experimental measurements and theoretical predictions for these observables are available in the future., Comment: 52 pages, 17 figures and 2 tables; More discussions and references added, final version appeared in JPG

There are four types of two-Higgs doublet models under a discrete $Z_2$ symmetry imposed to avoid tree-level flavour-changing neutral current, i.e. type-I, type-II, type-X and type-Y models. We investigate the possibility to discriminate the four models in the light of the flavour physics data, including $B_s-\bar B_s$ mixing, $B_{s,d} \to \mu^+ \mu^-$, $B\to \tau\nu$ and $\bar B \to X_s \gamma$ decays, the recent LHC Higgs data, the direct search for charged Higgs at LEP, and the constraints from perturbative unitarity and vacuum stability. After deriving the combined constraints on the Yukawa interaction parameters, we have shown that the correlation between the mass eigenstate rate asymmetry $A_{\Delta\Gamma}$ of $B_{s} \to \mu^+ \mu^-$ and the ratio $R={\cal B}(B_{s} \to \mu^+ \mu^-)_{exp}/ {\cal B}(B_{s} \to \mu^+ \mu^-)_{SM}$ could be sensitive probe to discriminate the four models with future precise measurements of the observables in the $B_{s} \to \mu^+ \mu^-$ decay at LHCb., Comment: 29 pages, 4 tables, 11 figures. v3: minor corrections included, matches published version in EPJC

Most recently, both BaBar and Belle experiments found evidences of neutral $D$ mixing. In this paper, we discuss the constraints on the strong phase difference in $D^0 \to K\pi$ decay from the measurements of the mixing parameters, $y^\prime$, $y_{CP}$ and $x$ at the $B$ factories. The sensitivity of the measurement of the mixing parameter $y$ is estimated in BES-III experiment at $\psi(3770)$ peak. We also make an estimate on the measurements of the mixing rate $R_M$. Finally, the sensitivity of the strong phase difference at BES-III are obtained by using data near the $D\bar{D}$ threshold with CP tag technique at BES-III experiment., Comment: 6 pages, 1 figure, a typo fixed, add more reference and a few corrections